Product decompositions of quasirandom groups and a Jordan type theorem

Nikolov, Nikolay, and Pyber, László. "Product decompositions of quasirandom groups and a Jordan type theorem." Journal of the European Mathematical Society 013.4 (2011): 1063-1077. <http://eudml.org/doc/277468>.

@article{Nikolov2011,
abstract = {We first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If $k$ is the minimal degree of a representation of the finite group $G$, then for every subset $B$ of $G$ with $|B|>|G|/k^\{1/3\}$ we have $B^3=G$. We use this to obtain improved versions of recent deep theorems of Helfgott and of Shalev concerning product decompositions of finite simple groups, with much simpler proofs. On the other hand, we prove a version of Jordan’s theorem which implies that if $k\ge 2$, then $G$ has a proper subgroup of index at most $c_0k^2$ for some constant $c_0$, hence a product-free subset of size at least $|G|/(ck)$. This answers a question of Gowers.},
author = {Nikolov, Nikolay, Pyber, László},
journal = {Journal of the European Mathematical Society},
keywords = {quasirandom groups; product-free sets; word values; finite simple groups; product-free subsets; product decompositions; word maps; character degrees; minimal degree representations},
language = {eng},
number = {4},
pages = {1063-1077},
publisher = {European Mathematical Society Publishing House},
title = {Product decompositions of quasirandom groups and a Jordan type theorem},
url = {http://eudml.org/doc/277468},
volume = {013},
year = {2011},
}

TY - JOUR
AU - Nikolov, Nikolay
AU - Pyber, László
TI - Product decompositions of quasirandom groups and a Jordan type theorem
JO - Journal of the European Mathematical Society
PY - 2011
PB - European Mathematical Society Publishing House
VL - 013
IS - 4
SP - 1063
EP - 1077
AB - We first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If $k$ is the minimal degree of a representation of the finite group $G$, then for every subset $B$ of $G$ with $|B|>|G|/k^{1/3}$ we have $B^3=G$. We use this to obtain improved versions of recent deep theorems of Helfgott and of Shalev concerning product decompositions of finite simple groups, with much simpler proofs. On the other hand, we prove a version of Jordan’s theorem which implies that if $k\ge 2$, then $G$ has a proper subgroup of index at most $c_0k^2$ for some constant $c_0$, hence a product-free subset of size at least $|G|/(ck)$. This answers a question of Gowers.
LA - eng
KW - quasirandom groups; product-free sets; word values; finite simple groups; product-free subsets; product decompositions; word maps; character degrees; minimal degree representations
UR - http://eudml.org/doc/277468
ER -